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  • S Ahlgren, K Ono
  • 2001
Eighty years ago, Ramanujan conjectured and proved some striking congruences for the partition function modulo powers of 5, 7, and 11. Until recently, only a handful of further such congruences were known. Here we report that such congruences are much more widespread than was previously known, and we describe the theoretical framework that appears to(More)
If p is prime, then let φ p denote the Legendre symbol modulo p and let p be the trivial character modulo p. « p be the Gaussian hypergeometric series over F p. For n > 2 the non-trivial values of n+1 F n (x) p have been difficult to obtain. Here we take the first step by obtaining a simple formula for 4 F 3 (1) p. As a corollary we obtain a result(More)
If F (z) is a newform of weight 2λ and D is a fundamental discriminant, then let L(F ⊗ χ D , s) be the usual twisted L-series. We study the algebraic parts of the central critical values of these twisted L-series modulo primes. We show that if there are two D (subject to some local conditions) for which the algebraic part of L(F ⊗ χ D , λ) is not 0 (mod),(More)